Howard Eves - Foundations and Fundamental Concepts of Mathematics

Following are the statements, in modern terminology, of the first twenty-eight propositions of Book I of Euclids Elements. The proofs of these propositions do not require the parallel postulate.

1. To construct an equilateral triangle on a given straight line segment.
2. From a given point to draw a straight line segment equal to a given straight line segment.
3. From the greater of two given straight line segments to cut off a part equal to the smaller.
4. Two triangles are congruent if two sides and the included angle of one are equal to two sides and the included angle of the other.
5. The base angles of an isosceles triangle are equal.
6. If two angles of a triangle are equal, the sides opposite these angles are equal.
7. . There cannot be two different triangles on the same side of a common base and having their other pairs of corresponding sides equal.
8. Two triangles are congruent if the three sides of one are equal to the three sides of the other.
9. To bisect a given angle.
10. To bisect a given straight line segment.
11. To erect a perpendicular to a given straight line at a given point on the line.
12. To drop a perpendicular from a given point to a given straight line.
13. Any pair of adjacent angles formed by two intersecting straight lines are either two right angles or are together equal to two right angles.
14. If, at any point on a straight line, two rays, on opposite sides of it, make a pair of adjacent angles together equal to two right angles, then these two rays lie in the same straight line.
15. Any pair of vertical angles formed by two intersecting straight lines are equal.
16. An exterior angle of a triangle is greater than either remote interior angle.
17. Any two angles of a triangle are together less than two right angles.
18. In a triangle the greater side is opposite the greater angle.
19. In a triangle the greater angle is opposite the greater side.
20. The sum of any two sides of a triangle is greater than the third side.
21. If two straight line segments are drawn from the ends of a side of a triangle to a point within the triangle, these two segments together will be less than the other two sides together but will contain a greater angle.
22. To construct a triangle having sides equal to three given straight line segments, any two of which are together greater than the third.
23. At a given point on a given straight line to construct an angle equal to a given angle.
24. If two triangles have two sides of one equal to two sides of the other, but the included angle of the first is greater than the included angle of the second, then the third side of the first is greater than the third side of the second.
25. If two triangles have two sides of one equal to two sides of the other, but the third side of the first is greater than the third side of the second, then the included angle of the first is greater than the included angle of the second.
26. Two triangles are congruent if two angles and a side of one are equal to two angles and the corresponding side of the other.
27. If a transversal of two straight lines makes a pair of alternate interior angles equal, then the two lines are parallel.
28. If a transversal of two straight lines makes a pair of corresponding angles equal, or a pair of interior angles on the same side of the transversal equal to two right angles, then the two lines are parallel.

We noted, in Section 2.3, that the first three of Euclids postulates state the primitive constructions from which all other constructions in the Elements are to be compounded. The first two of these postulates tell us what we can do with a Euclidean straightedge; we are permitted to draw as much as may be desired of the straight line determined by two given points. The third postulate tells us what we can do with Euclidean compasses; we are permitted to draw the circle of given center and passing through a given point. Note that neither instrument is to be used for transferring distances. This means that the straightedge cannot be marked, and the compasses must be regarded as having the characteristic that if one or both points be lifted from the paper, the instrument immediately collapses. For this reason, Euclidean compasses are often referred to as collapsing compasses ; they differ from modern compasses , which retain their opening and hence can be used as dividers for transferring distances. It would seem that modern compasses might be more powerful than the collapsing compasses. Curiously enough, such turns out not to be true; any construction performable with the modern compasses can also be carried out (in perhaps a longer way) by means of the collapsing compasses. We prove this fact as our first theorem.

Theorem 1 The collapsing and modern compasses are equivalent.

The circle with center O and passing through a given point C will be denoted by O ( C ), and the circle with center O and radius equal to a given segment AB will be denoted by O ( AB ). To prove the theorem it suffices to show that we may, with collapsing compasses, construct any circle O ( AB ). This may be accomplished as follows (see ). Draw circles A ( O ) and O ( A ) to intersect in D and E ; draw circles D ( B ) and E ( B ) to intersect again in F; draw circle O ( F ). It is an easy matter to prove that OF = AB , whence circle O ( F ) is circle O ( AB ).

Note: In view of Theorem 1, we may dispense with the Euclidean, or collapsing, compasses, and in their place employ the simpler modern compasses. We are assured that the set of constructions performable with straightedge and modern compasses is the same as the set performable with straightedge and Euclidean compasses.

We now proceed to establish a chain of theorems that will furnish us with a criterion for Euclidean constructibility. It turns out that the criterion is algebraic in nature. After the criterion is established, we shall apply it to prove that the three famous construction problems of antiquity (trisection of a general angle, duplication of a cube, and squaring of a circle) are impossible with Euclidean tools. A proof of this fact was not discovered until the nineteenth century, more than two thousand years after the original problems had been proposed.

Theorem 2 Given line segments of lengths a and b, and a unit segment, we can construct with Euclidean tools segments of lengths a + b, |ab|, ab, a/b, and a.

The first two constructions are trivial; the last three are apparent from .

Definition 1 Let a 1,... , a n be a given nonempty set of distinct real nonzero numbers, and let F 0 denote the set of all numbers each of which can be obtained from a 1,... , a n by a finite number of additions, subtractions, multiplications, and permissible divisions. Since F 0 is closed under addition, subtraction, multiplication, and permissible division, F 0 is a number field (see Section 5.1); it will be called the number field generated by a 1, ... , a n . The number field generated by the number 1 will be denoted by R 0.

Theorem 3 R0 is the field of all rational numbers. Every number field contains R0 as a subfield.

The proof is simple and is left to the reader.